Determine whether more general irreducible nonliftable \(PSU(2)\) instantons exist

Construct, or prove the nonexistence of, irreducible nonliftable \(PSU(2)\) instantons on Eguchi–Hanson space beyond the reducible Cartan-valued solutions captured by the cohomogeneity-one radial ansatz.

Background

Within the radial cohomogeneity-one ansatz, the paper finds genuine nonliftable PSU(2)PSU(2) configurations only on the reducible Cartan-valued branch. All smooth irreducible solutions in that ansatz have even flux parameter and lift to SU(2)SU(2).

The authors explicitly caution that this result is not a general nonexistence theorem. A broader construction would require working patchwise on a nontrivial SO(3)SO(3) bundle and imposing transition-function and bolt-regularity conditions not represented by the two global radial profile functions.

References

This conclusion applies to the particular cohomogeneity-one ansatz used here; it does not constitute a general nonexistence theorem for irreducible PSU(2) instantons on EH space. A more general nonliftable solution, if it exists, would have to be formulated patchwise on the nontrivial SO(3) bundle and would obey global transition and bolt regularity conditions not encoded by the two globally defined radial functions F(r) and G(r) alone.

— Gaugino condensates in $\mathcal{N}=1$ super-Yang-Mills Theory on Eguchi-Hanson space  (2610.00486 - Anber, 30 Sep 2026) in Section “Liftable and nonliftable solutions”